| import OAI.Combinatorics.HalvingLines.Noncollapse |
|
|
| namespace OAI |
|
|
| open Filter Topology |
|
|
| namespace PlanarHalving.Spatial |
| open Set Filter Metric |
| open scoped Topology |
|
|
| abbrev UnitSphere := {u : Point // ‖u‖=1} |
| instance : Nonempty UnitSphere := ⟨⟨(1,0),by simp [Prod.norm_def]⟩⟩ |
| instance : CompactSpace UnitSphere := by |
| have hc : IsCompact (sphere (0:Point) 1) := isCompact_sphere _ _ |
| have he : sphere (0:Point) 1 = {u | ‖u‖=1} := by ext x; simp |
| rw [he] at hc |
| exact isCompact_iff_compactSpace.mp hc |
| |
| noncomputable def radialU (f : Point → Point) (p : Point) (h : ℝ) : ℝ := |
| sSup (range fun u : UnitSphere => dist (f (p+h • (u:Point))) (f p)) |
| noncomputable def radialV (f : Point → Point) (p : Point) (h : ℝ) : ℝ := |
| sInf (range fun u : UnitSphere => dist (f (p+h • (u:Point))) (f p)) |
| |
| lemma unitSphere_param {p : Point} {h : ℝ} (hh : 0≤h) (u : UnitSphere) : |
| p+h • (u:Point)∈sphere p h := by |
| simp only [mem_sphere,dist_eq_norm,add_sub_cancel_left,norm_smul,Real.norm_eq_abs, |
| abs_of_nonneg hh,u.property,mul_one] |
| |
| lemma sphere_representation {p x : Point} {h : ℝ} (hh : 0<h) (hx : x∈sphere p h) : |
| ∃ u : UnitSphere, x=p+h • (u:Point) := by |
| have hn : ‖x-p‖=h := hx |
| refine ⟨⟨h⁻¹ • (x-p),?_⟩,?_⟩ |
| · rw [norm_smul,Real.norm_eq_abs,abs_of_pos (inv_pos.mpr hh),hn,inv_mul_cancel₀ hh.ne'] |
| · simp only [smul_smul,mul_inv_cancel₀ hh.ne',one_smul,add_sub_cancel] |
| |
| lemma continuous_sphereValue {f : Point → Point} {p : Point} {h : ℝ} (hh : 0≤h) |
| (hf : ContinuousOn f (closedBall p h)) : |
| Continuous (fun u : UnitSphere => dist (f (p+h • (u:Point))) (f p)) := by |
| have hc : Continuous (fun u : UnitSphere => p+h • (u:Point)) := by fun_prop |
| exact (hf.comp_continuous hc |
| (fun u => sphere_subset_closedBall (unitSphere_param (p:=p) hh u))).dist continuous_const |
| |
| lemma radialU_attained {f : Point → Point} {p : Point} {h : ℝ} (hh : 0<h) |
| (hf : ContinuousOn f (closedBall p h)) : |
| ∃ x ∈ sphere p h, dist (f x) (f p)=radialU f p h ∧ |
| ∀ y∈sphere p h, dist (f y) (f p)≤radialU f p h := by |
| obtain ⟨u,_,he,hu⟩ := isCompact_univ.exists_sSup_image_eq_and_ge |
| (Set.univ_nonempty : (univ:Set UnitSphere).Nonempty) (continuous_sphereValue hh.le hf).continuousOn |
| have he' : radialU f p h=dist (f (p+h • (u:Point))) (f p) := by |
| simpa only [image_univ,radialU] using he |
| refine ⟨_,unitSphere_param hh.le u,he'.symm,?_⟩ |
| intro y hy |
| obtain ⟨v,rfl⟩ := sphere_representation hh hy |
| rw [he'] |
| exact hu v (mem_univ _) |
| |
| lemma radialV_attained {f : Point → Point} {p : Point} {h : ℝ} (hh : 0<h) |
| (hf : ContinuousOn f (closedBall p h)) : |
| ∃ x ∈ sphere p h, dist (f x) (f p)=radialV f p h ∧ |
| ∀ y∈sphere p h, radialV f p h≤dist (f y) (f p) := by |
| obtain ⟨u,_,he,hu⟩ := isCompact_univ.exists_sInf_image_eq_and_le |
| (Set.univ_nonempty : (univ:Set UnitSphere).Nonempty) (continuous_sphereValue hh.le hf).continuousOn |
| have he' : radialV f p h=dist (f (p+h • (u:Point))) (f p) := by |
| simpa only [image_univ,radialV] using he |
| refine ⟨_,unitSphere_param hh.le u,he'.symm,?_⟩ |
| intro y hy |
| obtain ⟨v,rfl⟩ := sphere_representation hh hy |
| rw [he'] |
| exact hu v (mem_univ _) |
| |
| lemma radial_extrema_bounds {f : Point → Point} {p : Point} {h : ℝ} (hh : 0<h) |
| (hf : ContinuousOn f (closedBall p h)) : |
| 0≤radialV f p h ∧ radialV f p h≤radialU f p h ∧ |
| ∀ x∈sphere p h, radialV f p h≤dist (f x) (f p) ∧ dist (f x) (f p)≤radialU f p h := by |
| obtain ⟨x,hx,he,hlo⟩ := radialV_attained hh hf |
| obtain ⟨y,hy,he',hup⟩ := radialU_attained hh hf |
| refine ⟨he ▸ dist_nonneg,?_,fun z hz => ⟨hlo z hz,hup z hz⟩⟩ |
| rw [← he']; exact hlo y hy |
| |
| lemma radialV_pos {f : Point → Point} {p : Point} {h : ℝ} (hh : 0<h) |
| (hf : ContinuousOn f (closedBall p h)) (hinj : InjOn f (closedBall p h)) : |
| 0<radialV f p h := by |
| obtain ⟨x,hx,he,_⟩ := radialV_attained hh hf |
| rw [← he,dist_pos] |
| intro hf' |
| have hxp := hinj (sphere_subset_closedBall hx) (mem_closedBall_self hh.le) hf' |
| have hh' : h=0 := by simpa only [hxp,dist_self,eq_comm,mem_sphere] using hx |
| exact hh.ne' hh' |
| |
| lemma continuousOn_radialU {f : Point → Point} {D P : Set Point} {h : ℝ} (hh : 0≤h) |
| (hf : ContinuousOn f D) (hP : ∀ p∈P, closedBall p h ⊆ D) : |
| ContinuousOn (fun p => radialU f p h) P := by |
| rw [continuousOn_iff_continuous_domRestrict] |
| let a : P × UnitSphere → Point := fun q => (q.1:Point)+h • (q.2:Point) |
| have ha : Continuous a := by dsimp [a]; fun_prop |
| have hfa : Continuous (f ∘ a) := hf.comp_continuous ha |
| (fun q => hP q.1 q.1.property (sphere_subset_closedBall (unitSphere_param hh q.2))) |
| have hfb : Continuous (fun q : P × UnitSphere => f q.1) := hf.comp_continuous |
| (continuous_subtype_val.comp continuous_fst) (fun q => hP q.1 q.1.property (mem_closedBall_self hh)) |
| have ht := isCompact_univ.continuous_sSup |
| (f:=fun p : P => fun u : UnitSphere => dist (f ((p:Point)+h • (u:Point))) (f p)) (hfa.dist hfb) |
| change Continuous (fun p : P => sSup (range fun u : UnitSphere => dist (f ((p:Point)+h • (u:Point))) (f p))) |
| simpa only [image_univ] using ht |
| |
| lemma continuousOn_radialV {f : Point → Point} {D P : Set Point} {h : ℝ} (hh : 0≤h) |
| (hf : ContinuousOn f D) (hP : ∀ p∈P, closedBall p h ⊆ D) : |
| ContinuousOn (fun p => radialV f p h) P := by |
| rw [continuousOn_iff_continuous_domRestrict] |
| let a : P × UnitSphere → Point := fun q => (q.1:Point)+h • (q.2:Point) |
| have ha : Continuous a := by dsimp [a]; fun_prop |
| have hfa : Continuous (f ∘ a) := hf.comp_continuous ha |
| (fun q => hP q.1 q.1.property (sphere_subset_closedBall (unitSphere_param hh q.2))) |
| have hfb : Continuous (fun q : P × UnitSphere => f q.1) := hf.comp_continuous |
| (continuous_subtype_val.comp continuous_fst) (fun q => hP q.1 q.1.property (mem_closedBall_self hh)) |
| have ht := isCompact_univ.continuous_sInf |
| (f:=fun p : P => fun u : UnitSphere => dist (f ((p:Point)+h • (u:Point))) (f p)) (hfa.dist hfb) |
| change Continuous (fun p : P => sInf (range fun u : UnitSphere => dist (f ((p:Point)+h • (u:Point))) (f p))) |
| simpa only [image_univ] using ht |
| |
| end PlanarHalving.Spatial |
| |
| namespace PlanarHalving.Spatial |
| open Set Filter MeasureTheory Metric Zoom |
| open scoped Topology |
| |
| noncomputable def rankCore (h : ℝ) : Set Point := |
| Icc (2*h) (1-2*h) ×ˢ Icc (-1+2*h) (-2*h) |
| |
| lemma rankCore_closed (h : ℝ) : IsClosed (rankCore h) := isClosed_Icc.prod isClosed_Icc |
| |
| lemma rankCore_ball {h : ℝ} (hh : 0<h) {p : Point} (hp : p∈rankCore h) : |
| closedBall p h ⊆ Ioo (0:ℝ) 1 ×ˢ Ioo (-1:ℝ) 0 := by |
| intro x hx |
| rw [mem_closedBall,Prod.dist_eq] at hx |
| have hx1 : |x.1-p.1|≤h := (le_max_left _ _).trans hx |
| have hx2 : |x.2-p.2|≤h := (le_max_right _ _).trans hx |
| rcases abs_le.mp hx1 with ⟨hx11,hx12⟩ |
| rcases abs_le.mp hx2 with ⟨hx21,hx22⟩ |
| rcases hp with ⟨⟨hp11,hp12⟩,⟨hp21,hp22⟩⟩ |
| exact ⟨⟨by linarith,by linarith⟩,⟨by linarith,by linarith⟩⟩ |
| |
| lemma rankCore_small {p : Point} (hp : p∈Ioo (0:ℝ) 1 ×ˢ Ioo (-1:ℝ) 0) : |
| ∃ δ>0, ∀ h, h≤δ → p∈rankCore h := by |
| let d := min p.1 (min (1-p.1) (min (p.2+1) (-p.2))) |
| have hd : 0<d := by dsimp [d]; exact lt_min hp.1.1 (lt_min (by linarith [hp.1.2]) (lt_min (by linarith [hp.2.1]) (by linarith [hp.2.2]))) |
| have h1 : d≤p.1 := min_le_left _ _ |
| have h2 : d≤1-p.1 := (min_le_right _ _).trans (min_le_left _ _) |
| have h3 : d≤p.2+1 := (min_le_right _ _).trans ((min_le_right _ _).trans (min_le_left _ _)) |
| have h4 : d≤-p.2 := (min_le_right _ _).trans ((min_le_right _ _).trans (min_le_right _ _)) |
| refine ⟨d/4,by positivity,fun h hh => ?_⟩ |
| exact ⟨⟨by linarith,by linarith⟩,⟨by linarith,by linarith⟩⟩ |
| |
| lemma radialCompare_extrema {f : Point → Point} {p : Point} {h C M T : ℝ} |
| (hh : 0<h) (hT : 1≤T) (hM : 0<M) (hCM : C≤M) |
| (hf : ContinuousOn f (closedBall p h)) (hc : RadialCompare f p h C T) : |
| (1/(M*T))*radialU f p h ≤ radialV f p (h/T) ∧ |
| radialU f p (h/T)≤(M/T)*radialV f p h := by |
| have hTp : 0<T := lt_of_lt_of_le zero_lt_one hT |
| have hhT : 0<h/T := div_pos hh hTp |
| have hsmall : h/T≤h := (div_le_iff₀ hTp).mpr (by nlinarith) |
| have hf' := hf.mono (closedBall_subset_closedBall hsmall) |
| obtain ⟨u,hu,heu,_⟩ := radialV_attained hhT hf' |
| obtain ⟨v,hv,hev,_⟩ := radialU_attained hh hf |
| have h1 := hc.1 u v hu hv |
| change dist (f v) (f p) ≤ C*T*dist (f u) (f p) at h1 |
| rw [heu,hev] at h1 |
| have hlow : 0≤radialV f p (h/T) := (radial_extrema_bounds hhT hf').1 |
| have h1' : radialU f p h ≤ M*T*radialV f p (h/T) := |
| h1.trans (mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hCM hTp.le) hlow) |
| obtain ⟨u',hu',heu',_⟩ := radialU_attained hhT hf' |
| obtain ⟨v',hv',hev',_⟩ := radialV_attained hh hf |
| have h2 := hc.2 u' v' hu' hv' |
| change T*dist (f u') (f p) ≤ C*dist (f v') (f p) at h2 |
| rw [heu',hev'] at h2 |
| have hv0 := (radial_extrema_bounds hh hf).1 |
| constructor |
| · rw [one_div,inv_mul_eq_div] |
| exact (div_le_iff₀ (mul_pos hM hTp)).mpr (by nlinarith [h1']) |
| · rw [div_mul_eq_mul_div] |
| apply (le_div_iff₀ hTp).mpr |
| have h2' := h2.trans (mul_le_mul_of_nonneg_right hCM hv0) |
| convert h2' using 1; ring |
| |
| def ScaleGood (f : ℕ → Point → Point) (h q α β : ℝ) : Set Point := |
| {p | p∈rankCore h ∧ ∀ j : ℕ, ∀ᶠ n in atTop, |
| α*radialU (f n) p (h*q^j)≤radialV (f n) p (h*q^(j+1)) ∧ |
| radialU (f n) p (h*q^(j+1))≤β*radialV (f n) p (h*q^j)} |
| |
| lemma scaleGood_measurable (f : ℕ → Point → Point) {h q α β : ℝ} |
| (hh : 0<h) (hq : 0≤q) (hq1 : q≤1) |
| (hf : ∀ n, ContinuousOn (f n) (Ioo (0:ℝ) 1 ×ˢ Ioo (-1:ℝ) 0)) : |
| MeasurableSet (ScaleGood f h q α β) := by |
| let P := rankCore h |
| have hcU (n j : ℕ) : Continuous (fun p : P => radialU (f n) p (h*q^j)) := by |
| have hc : ContinuousOn (fun p => radialU (f n) p (h*q^j)) P := by |
| apply continuousOn_radialU (by positivity) (hf n) |
| intro p hp |
| exact (closedBall_subset_closedBall (mul_le_of_le_one_right hh.le (pow_le_one₀ hq hq1))).trans |
| (rankCore_ball hh hp) |
| exact hc.domRestrict |
| have hcV (n j : ℕ) : Continuous (fun p : P => radialV (f n) p (h*q^j)) := by |
| have hc : ContinuousOn (fun p => radialV (f n) p (h*q^j)) P := by |
| apply continuousOn_radialV (by positivity) (hf n) |
| intro p hp |
| exact (closedBall_subset_closedBall (mul_le_of_le_one_right hh.le (pow_le_one₀ hq hq1))).trans |
| (rankCore_ball hh hp) |
| exact hc.domRestrict |
| have hm : MeasurableSet {p : P | ∀ j : ℕ, ∀ᶠ n in atTop, |
| α*radialU (f n) p (h*q^j)≤radialV (f n) p (h*q^(j+1)) ∧ |
| radialU (f n) p (h*q^(j+1))≤β*radialV (f n) p (h*q^j)} := by |
| simp only [eventually_atTop] |
| rw [ofPred_forall] |
| apply MeasurableSet.iInter |
| intro j |
| rw [ofPred_exists] |
| apply MeasurableSet.iUnion |
| intro N |
| rw [ofPred_forall] |
| apply MeasurableSet.iInter |
| intro n |
| by_cases hn : N≤n |
| · simp only [hn,true_implies] |
| exact (measurableSet_le ((hcU n j).measurable.const_mul α) (hcV n (j+1)).measurable).inter |
| (measurableSet_le (hcU n (j+1)).measurable ((hcV n j).measurable.const_mul β)) |
| · simp only [hn,false_implies,ofPred_true]; exact MeasurableSet.univ |
| have hi := (MeasurableEmbedding.subtype_coe (rankCore_closed h).measurableSet).measurableSet_image.mpr hm |
| convert hi using 1 |
| ext p |
| simp only [ScaleGood,mem_ofPred_eq,mem_image,Subtype.exists] |
| aesop |
| |
| end PlanarHalving.Spatial |
| |
| namespace PlanarHalving.Spatial |
| open Set Filter MeasureTheory Metric Zoom |
| open scoped Topology |
| |
| lemma exists_scaleGood (f : ℕ → Point → Point) {p : Point} |
| (hp : p∈Ioo (0:ℝ) 1 ×ˢ Ioo (-1:ℝ) 0) |
| (hf : ∀ n, ContinuousOn (f n) (Ioo (0:ℝ) 1 ×ˢ Ioo (-1:ℝ) 0)) |
| {C M T : ℝ} (hCM : C≤M) (hM : 0<M) (hT : 1<T) |
| (hc : ∀ᶠ h in 𝓝[>] (0:ℝ), ∀ᶠ n in atTop, RadialCompare (f n) p h C T) : |
| ∃ m : ℕ, p∈ScaleGood f ((1/T)^m) (1/T) (1/(M*T)) (M/T) := by |
| obtain ⟨δ,hδ,hδc⟩ := mem_nhdsGT_iff_exists_Ioo_subset.mp hc |
| obtain ⟨d,hd,hcore⟩ := rankCore_small hp |
| have hTp : 0<T := lt_trans zero_lt_one hT |
| have hq : 0<1/T := div_pos zero_lt_one hTp |
| have hq1 : 1/T<1 := (div_lt_one hTp).mpr hT |
| have hlim := tendsto_pow_atTop_nhds_zero_of_lt_one hq.le hq1 |
| obtain ⟨m,hm⟩ := ((tendsto_order.mp hlim).2 (min δ d) (lt_min hδ hd)).exists |
| have hmd : (1/T)^m<d := hm.trans_le (min_le_right _ _) |
| have hmδ : (1/T)^m<δ := hm.trans_le (min_le_left _ _) |
| have hmc := hcore _ hmd.le |
| refine ⟨m,hmc,fun j => ?_⟩ |
| have hr : 0<(1/T)^m*(1/T)^j := mul_pos (pow_pos hq _) (pow_pos hq _) |
| have hrm : (1/T)^m*(1/T)^j≤(1/T)^m := mul_le_of_le_one_right |
| (pow_nonneg hq.le _) (pow_le_one₀ hq.le hq1.le) |
| have hrc := hδc ⟨hr,hrm.trans_lt hmδ⟩ |
| filter_upwards [hrc] with n hn |
| have hball := (closedBall_subset_closedBall hrm (x:=p)).trans |
| (rankCore_ball (pow_pos hq _) hmc) |
| have hb := radialCompare_extrema hr hT.le hM hCM ((hf n).mono hball) hn |
| have he : (1/T)^m*(1/T)^j/T=(1/T)^m*(1/T)^(j+1) := by rw [pow_succ]; ring |
| rwa [he] at hb |
| |
| lemma exists_dyadicExponent_scaleGood (f : ℕ → Point → Point) {p : Point} |
| (hp : p∈Ioo (0:ℝ) 1 ×ˢ Ioo (-1:ℝ) 0) |
| (hf : ∀ n, ContinuousOn (f n) (Ioo (0:ℝ) 1 ×ˢ Ioo (-1:ℝ) 0)) |
| (hc : ∃ C : ℝ, 0<C ∧ ∀ T : ℝ, 1≤T → ∀ᶠ h in 𝓝[>] (0:ℝ), |
| ∀ᶠ n in atTop, RadialCompare (f n) p h C T) : |
| ∃ M m : ℕ, 2≤M ∧ p∈ScaleGood f ((1/(M:ℝ)^16)^m) (1/(M:ℝ)^16) |
| (1/((M:ℝ)*(M:ℝ)^16)) ((M:ℝ)/(M:ℝ)^16) := by |
| obtain ⟨C,_,hC⟩ := hc |
| obtain ⟨N,hN⟩ := exists_nat_gt C |
| let M := max N 2 |
| have hM2 : 2≤M := le_max_right _ _ |
| have hMp : 0<(M:ℝ) := by exact_mod_cast (lt_of_lt_of_le (by decide : 0<2) hM2) |
| have hM1 : 1<(M:ℝ) := by exact_mod_cast (lt_of_lt_of_le (by decide : 1<2) hM2) |
| have hCM : C≤(M:ℝ) := hN.le.trans (by exact_mod_cast le_max_left N 2) |
| have hT : 1<(M:ℝ)^16 := one_lt_pow₀ hM1 (by decide) |
| obtain ⟨m,hm⟩ := exists_scaleGood f hp hf hCM hMp hT (hC _ hT.le) |
| exact ⟨M,m,hM2,hm⟩ |
| |
| lemma positive_scaleGood {X : Type*} [MeasurableSpace X] |
| (μ : Measure X) (E : Set X) (hE : MeasurableSet E) (hpos : μ E≠0) |
| (π : X → Point) (hπ : Measurable π) (f : ℕ → Point → Point) |
| (hf : ∀ n, ContinuousOn (f n) (Ioo (0:ℝ) 1 ×ˢ Ioo (-1:ℝ) 0)) |
| (hc : ∀ᵐ x ∂μ.restrict E, π x∈Ioo (0:ℝ) 1 ×ˢ Ioo (-1:ℝ) 0 ∧ |
| ∃ C : ℝ, 0<C ∧ ∀ T : ℝ, 1≤T → ∀ᶠ h in 𝓝[>] (0:ℝ), |
| ∀ᶠ n in atTop, RadialCompare (f n) (π x) h C T) : |
| ∃ M m : ℕ, 2≤M ∧ MeasurableSet (E∩π ⁻¹' ScaleGood f ((1/(M:ℝ)^16)^m) |
| (1/(M:ℝ)^16) (1/((M:ℝ)*(M:ℝ)^16)) ((M:ℝ)/(M:ℝ)^16)) ∧ |
| μ (E∩π ⁻¹' ScaleGood f ((1/(M:ℝ)^16)^m) |
| (1/(M:ℝ)^16) (1/((M:ℝ)*(M:ℝ)^16)) ((M:ℝ)/(M:ℝ)^16))≠0 := by |
| let A := fun q : ℕ×ℕ => {x | 2≤q.1 ∧ π x∈ScaleGood f ((1/(q.1:ℝ)^16)^q.2) |
| (1/(q.1:ℝ)^16) (1/((q.1:ℝ)*(q.1:ℝ)^16)) ((q.1:ℝ)/(q.1:ℝ)^16)} |
| obtain ⟨q,hq⟩ := Limits.positive_countable_selection μ E hE hpos A (by |
| filter_upwards [hc] with x hx |
| obtain ⟨M,m,hM,hm⟩ := exists_dyadicExponent_scaleGood f hx.1 hf hx.2 |
| exact ⟨(M,m),hM,hm⟩) |
| obtain ⟨x,_,hM,_⟩ := nonempty_of_measure_ne_zero hq |
| have hM2 : 2≤q.1 := hM |
| have hMp : 0<(q.1:ℝ) := by exact_mod_cast (lt_of_lt_of_le (by decide : 0<2) hM2) |
| have hM1 : 1≤(q.1:ℝ) := by exact_mod_cast (le_trans (by decide : 1≤2) hM2) |
| have hT : 1≤(q.1:ℝ)^16 := one_le_pow₀ hM1 |
| have hTp : 0<(q.1:ℝ)^16 := pow_pos hMp _ |
| have hm := scaleGood_measurable f (h:=(1/(q.1:ℝ)^16)^q.2) |
| (α:=1/((q.1:ℝ)*(q.1:ℝ)^16)) (β:=(q.1:ℝ)/(q.1:ℝ)^16) |
| (pow_pos (div_pos zero_lt_one hTp) q.2) |
| (div_nonneg zero_le_one hTp.le) ((div_le_one hTp).mpr hT) hf |
| refine ⟨q.1,q.2,hM2,hE.inter (hm.preimage hπ),?_⟩ |
| exact ne_of_gt ((pos_iff_ne_zero.mpr hq).trans_le (measure_mono (by |
| intro x hx |
| exact ⟨hx.1,hx.2.2⟩))) |
| |
| end PlanarHalving.Spatial |
| |
| namespace PlanarHalving.Spatial |
| open Set Filter Metric |
| open scoped Topology |
| |
| lemma radius_le_of_ball_subset_closedBall {a b : Point} {r R : ℝ} |
| (hr : 0≤r) (hR : 0≤R) (hb : ball a r ⊆ closedBall b R) : r≤R := by |
| have hh := diam_mono hb isBounded_closedBall |
| rw [diam_ball_eq a hr,diam_closedBall_eq b hR] at hh |
| linarith |
| |
| lemma normalized_local_bounds (f : OpenPartialHomeomorph Point Point) |
| (hs : f.source=univ) {p₀ p : Point} {h q : ℝ} |
| (hh : 0<h) (hq : 0<q) (hq2 : q≤1/2) |
| (hb₀ : closedBall p₀ h ⊆ f.target) (hb : closedBall p h ⊆ f.target) |
| (hp : dist p p₀<h/4) (hz : f.symm p₀=0) (hU : radialU f.symm p₀ h=1) : |
| ‖f.symm p‖≤1 ∧ radialU f.symm p (h*q)≤2 ∧ |
| ∀ α : ℝ, 0≤α → α≤radialV f.symm p₀ (h*q) → α≤radialU f.symm p h := by |
| have hqh : h*q≤h := mul_le_of_le_one_right hh.le (by linarith) |
| have hqhp : 0<h*q := mul_pos hh hq |
| have hcont₀ := f.continuousOn_symm.mono hb₀ |
| have hcont := f.continuousOn_symm.mono hb |
| have hcont₁ := hcont.mono (closedBall_subset_closedBall hqh) |
| have hcont₀₁ := hcont₀.mono (closedBall_subset_closedBall hqh) |
| have hby : closedBall p (h*q) ⊆ closedBall p₀ h := by |
| intro x hx |
| have ht := dist_triangle x p p₀ |
| have hx' : dist x p≤h*q := hx |
| have hm : h*q≤h/2 := by nlinarith |
| change dist x p₀≤h |
| linarith |
| have hby' : ball p₀ (h*q) ⊆ closedBall p h := by |
| intro x hx |
| have ht := dist_triangle x p₀ p |
| have hx' : dist x p₀<h*q := hx |
| rw [dist_comm p₀ p] at ht |
| have hm : h*q≤h/2 := by nlinarith |
| change dist x p≤h |
| linarith |
| have hupper : ∀ x∈closedBall p₀ h, ‖f.symm x‖≤1 := by |
| have hr := radial_upper_enclosure f hs (hU:=zero_le_one) hb₀ (fun x hx => by |
| have hhx := (radial_extrema_bounds hh hcont₀).2.2 x hx |
| simpa only [hU] using hhx.2) |
| simpa only [hz,dist_zero_right] using hr |
| have hnp : ‖f.symm p‖≤1 := hupper p (by change dist p p₀≤h; linarith) |
| refine ⟨hnp,?_,?_⟩ |
| · obtain ⟨x,hx,he,_⟩ := radialU_attained hqhp hcont₁ |
| have hnx := hupper x (hby (sphere_subset_closedBall hx)) |
| rw [← he] |
| rw [dist_eq_norm] |
| exact (norm_sub_le _ _).trans (by linarith) |
| · intro α hα hαV |
| have hinner := radial_inner_enclosure f hs (hb₀ (mem_closedBall_self hh.le)) |
| hqhp.le (fun x hx => hαV.trans ((radial_extrema_bounds hqhp hcont₀₁).2.2 x hx).1) |
| rw [hz] at hinner |
| have hu0 := (radial_extrema_bounds hh hcont).1.trans (radial_extrema_bounds hh hcont).2.1 |
| apply radius_le_of_ball_subset_closedBall hα hu0 |
| intro y hy |
| obtain ⟨x,hx,rfl⟩ := hinner hy |
| exact radial_upper_enclosure f hs hu0 hb |
| (fun x hx => ((radial_extrema_bounds hh hcont).2.2 x hx).2) x (hby' hx) |
| |
| lemma scaleGood_iteration (f : ℕ → OpenPartialHomeomorph Point Point) |
| (hs : ∀ n, (f n).source=univ) |
| (ht : ∀ n, (f n).target=Ioo (0:ℝ) 1 ×ˢ Ioo (-1:ℝ) 0) |
| {p₀ p : Point} {h q α β : ℝ} (hh : 0<h) (hq : 0<q) (hq2 : q≤1/2) |
| (hα : 0≤α) (hβ : 0≤β) |
| (h₀ : p₀∈ScaleGood (fun n => (f n).symm) h q α β) |
| (hp : p∈ScaleGood (fun n => (f n).symm) h q α β) (hd : dist p p₀<h/4) |
| (hz : ∀ n, (f n).symm p₀=0) (hU : ∀ n, radialU (f n).symm p₀ h=1) : |
| (∀ n, ‖(f n).symm p‖≤1) ∧ ∀ j : ℕ, ∀ᶠ n in atTop, |
| α^(j+2)≤radialV (f n).symm p (h*q^(j+1)) ∧ |
| radialU (f n).symm p (h*q^(j+1))≤2*β^j := by |
| have hb (n : ℕ) : closedBall p h ⊆ (f n).target := by rw [ht n]; exact rankCore_ball hh hp.1 |
| have hb₀ (n : ℕ) : closedBall p₀ h ⊆ (f n).target := by rw [ht n]; exact rankCore_ball hh h₀.1 |
| have hlocal := fun n => normalized_local_bounds (f n) (hs n) hh hq hq2 (hb₀ n) (hb n) hd (hz n) (hU n) |
| refine ⟨fun n => (hlocal n).1,?_⟩ |
| have hlow : ∀ᶠ n in atTop, α≤radialU (f n).symm p h := by |
| filter_upwards [h₀.2 0] with n hn |
| apply (hlocal n).2.2 α hα |
| simpa only [Nat.zero_add,pow_zero,pow_one,mul_one,hU n] using hn.1 |
| intro j |
| induction j with |
| | zero => |
| filter_upwards [hlow,hp.2 0] with n hn hn' |
| constructor |
| · have hh' := mul_le_mul_of_nonneg_left hn hα |
| simp only [Nat.zero_add,pow_zero,pow_one,mul_one] at hn' |
| simpa only [Nat.zero_add,pow_one,pow_two] using hh'.trans hn'.1 |
| · simpa only [Nat.zero_add,pow_zero,pow_one,mul_one] using (hlocal n).2.1 |
| | succ j ih => |
| filter_upwards [ih,hp.2 (j+1)] with n hn hn' |
| have hjh : h*q^(j+1)≤h := mul_le_of_le_one_right hh.le (pow_le_one₀ hq.le (by linarith)) |
| have hc := (f n).continuousOn_symm.mono ((closedBall_subset_closedBall hjh).trans (hb n)) |
| have hvu := (radial_extrema_bounds (mul_pos hh (pow_pos hq _)) hc).2.1 |
| constructor |
| · have he := (mul_le_mul_of_nonneg_left (hn.1.trans hvu) hα).trans hn'.1 |
| calc |
| α^(j.succ+2) = α*α^(j+2) := by |
| rw [show j.succ+2=j+2+1 by omega,pow_succ,mul_comm] |
| _ ≤ _ := he |
| · have he := hn'.2.trans (mul_le_mul_of_nonneg_left (hvu.trans hn.2) hβ) |
| calc |
| _ ≤ β*(2*β^j) := he |
| _ = 2*β^j.succ := by rw [pow_succ]; ring |
| |
| end PlanarHalving.Spatial |
| |
| end OAI |
| |
